College Trigonometry, 2011a

10.5 Graphs of the Trigonometric Functions 805
From the graph, it appears as if the tangent function is periodic with period π. To prove that this
is the case, we appeal to the sum formula for tangents. We have:
tan(x + π) = tan(x)+tan(π)
1 − tan(x) tan(π) = tan(x)+0
1 − (tan(x))(0) = tan(x),
which tells us the period of tan(x) isatmostπ. To show that it is exactly π, supposep is a
positive real number so that tan(x + p) =tan(x) for all real numbers x. For x =0,wehave
tan(p) =tan(0+p) = tan(0) = 0, which means p is a multiple of π. The smallest positive multiple
of π is π itself, so we have established the result. We take as our fundamental cycle for y = tan(x)
the interval ( − π 2 , π )
2 , and use as our ‘quarter marks’ x = −
π
2 , − π 4 ,0, π 4 and π 2
. From the graph, we
see confirmation of our domain and range work in Section 10.3.1.
It should be no surprise that K(x) =cot(x) behaves similarly to J(x) =tan(x). Plotting cot(x)
over the interval [0, 2π] results in the graph below.
y
x cot(x) (x, cot(x))
0 undefined
π
4
1
π
2
0
3π
4
−1
π undefined
5π
4
1
3π
2
0
7π
4
−1
2π undefined
( π
4
( , 1)
π
2
( , 0)
3π
4 , −1)
( 5π
4
( , 1)
3π
2
( , 0)
7π
4 , −1)
1
−1
π
4
π
2
3π
4
π
5π
4
3π
2
7π
4
2π
x
The graph of y = cot(x) over[0, 2π].
From these data, it clearly appears as if the period of cot(x) isπ, and we leave it to the reader
to prove this. 14 We take as one fundamental cycle the interval (0,π) with quarter marks: x =0,
π
4 , π 2 , 3π 4
and π. A more complete graph of y = cot(x) is below, along with the fundamental cycle
highlighted as usual. Once again, we see the domain and range of K(x) =cot(x) asreadfromthe
graph matches with what we found analytically in Section 10.3.1.
14 Certainly, mimicking the proof that the period of tan(x) is an option; for another approach, consider transforming
tan(x) tocot(x) using identities.